I expect significance level to be countable, so without the my ear expects a. I expect significance to be uncountable, so it works without a. But I can't make any comment on whether the NPs in either question should be definite or indefinite because it depends on context you haven't included.
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snailboatMay 1 '14 at 1:39

I've added a complete sentence. Does that clarify the context? Thank you four your help!
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user1357015May 1 '14 at 1:40

I read it as the groups and the level, which to my ear sound correct.
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medicaMay 1 '14 at 1:42

@medica, so the "the" is needed in both cases. What happens if you take out "level", what about "At 5% significance..."
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user1357015May 1 '14 at 1:44

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I don't like statistics, so I'll use something similar: blood alcohol levels. I would say, At .12%, the patient's blood alcohol exceeds the legal limit... but at the .12% level, he is too impaired... Where the first makes a percentage the object of the preposition at, and the second makes level the object of the preposition (if that helps). So, I could live with At 5%...
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medicaMay 1 '14 at 1:50

1 Answer
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For 1. You need the article. Without an article, quantification in English is 'For all' or 'In general'. You are not comparing all red, green, and blue groups, everywhere, across all time. You are only comparing the ones relevant to this study. So you need an article.

For 2. You need the article. Again, no article means 'In general'. If you left it off, you would again be taking about all 5% significance levels, in general. That would at least be plural. Besides, you don't intend to discuss all, or even multiple, 5% significance levels, just the one you computed.

If you take the word 'level' out, you are breaking the idiom, and you need a different preposition. One can be 'at a level', but not 'at a significance'. The metaphor of a point on the measuring-stick as a place is lost.

You can say "With 5% significance or better" (not higher, lower, but that means better). But almost nobody does, because the field of statistics comes packaged with a set of formulaic idioms that all scientists usually stick with.

The more specific reason nobody varies from the narrow idiom, is that this choice of vocabulary does not stand up well to casual manipulation. Mostly because it really does not make a lick of sense. But it is traditional.

Everyone wants more significant results, but the number that represents significance is defined to be an estimate of the chance your data has misled you. So higher significance, in the natural language sense, means this number must be lower.

Also, you do not know the value of that number. You never computed it, you did computations to find out what values other numbers must have in order to ensure that that number was at most 5%. It may not actually have a value near 5%. It could be much lower.

But if it actually were lower, finding that out and publishing it would be unethical. Because it would imply you had either done this the hard way, with full intent to publish the exact confidence range, or decided ahead of time not to publish unless you got stellar results. And that would misrepresent your approach, and make your work hard to reproduce.

So it is just unwise to mess with the rigid formulaic way of talking about this particular statistic. The best thing to do is to not take the first step toward colloquialism, and just use the phrasing that the community has frozen in place, when you are writing.

At 5% significance is OK. But not *at 5% significance level; with level one needs the article.
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John LawlerMay 1 '14 at 2:40

It may be accepted, but is an abuse of convention. People fill in what you left out, but you are really violating the idiom. Your risk is not a measure. Although you might say 'At the level of risk where I have a 5% chance of being wrong', you should not say 'At a 5% chance I am wrong', you should say 'With a 5% chance I am wrong'. Diluting the metaphor weakens the mental model.
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Jon Jay ObermarkMay 1 '14 at 2:52

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Idioms are not required by law, and may be used, or not, by any speaker. Do you have any references for these claims?
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John LawlerMay 1 '14 at 3:02

Sorry, my response was too hot. I put the relevant argument into the main post. But really, it is "At the 5% level of significance" that is the idiom. What you really mean is much harder to say. So respect the form of the idiom. Especially since it comes dangerously close to making no sense at all.
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Jon Jay ObermarkMay 1 '14 at 4:29

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It doesn't make any sense at all to non-scientists. You have to understand what "significance" means, and that takes basic statistics and experimental design knowledge. If you have to understand all that already, what's a predictable noun doing for you except upholding your dignity? If you're afraid your pants will fall down if you don't respect the idiom, you're putting way too much scientific faith in form instead of substance. Scientists talk the same way everybody else does, and they don't respect idioms any more than other people, unless it's a matter of turf.
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John LawlerMay 1 '14 at 15:21